Discussion
The remainder when the positive integer m is divided by 7 is x. The remainder when m is divided by 14 is x+7.
Which one of the following could m equal?
Answer: B
Test each candidate against both conditions, since the second is far more restrictive than it looks:
| m | m mod 7 = x | m mod 14 | x + 7 | fits? |
|---|---|---|---|---|
| 45 | 3 | 3 | 10 | no |
| 53 | 4 | 11 | 11 | yes |
| 72 | 2 | 2 | 9 | no |
| 85 | 1 | 1 | 8 | no |
The pattern behind it: dividing by 14 leaves either \( x \) or \( x+7 \), and the second happens exactly when the quotient by 7 is odd. Here \( 53 = 7 \times 7 + 4 \), and that quotient 7 is odd, so the remainder on 14 is \( 4 + 7 = 11 \).
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